Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Limiting distributions of theta series on Siegel half-spaces

Authors: F. Götze and M. Gordin
Translated by:
Original publication: Algebra i Analiz, tom 15 (2003), nomer 1.
Journal: St. Petersburg Math. J. 15 (2004), 81-102
MSC (2000): Primary 11Fxx, 37D30
Published electronically: December 31, 2003
MathSciNet review: 1979719
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $m \ge 1$ be an integer. For any $Z$ in the Siegel upper half-space we consider the multivariate theta series

\begin{displaymath}\Theta (Z)= \sum _{{\overline{n}} \in \mathbb{Z}^{m}} \exp (\pi i \,{}^{t} {\overline{n}} Z {\overline{n}}).\end{displaymath}

The function $\Theta $ is invariant with respect to every substitution $Z\longmapsto Z + P$, where $P$ is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix $Y > 0$ the function $\Theta _{Y} ( \cdot ) = (\det Y)^{1/4} \Theta (\cdot +iY)$ can be viewed as a complex-valued random variable on the torus $\mathbb{T}^{m(m+1)/2}$ with the Haar probability measure. It is proved that the weak limit of the distribution of $ \Theta _{\tau Y}$ as $\tau \to 0$ exists and does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of Dani-Margulis' and Ratner's results on dynamics of unipotent flows.

References [Enhancements On Off] (What's this?)

  • [Ar] V. I. Arnol'd, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR 138 (1961), no. 2, 255-257; English transl. in Soviet Math. Dokl. 2 (1961). MR 28:1555
  • [Da] S. G. Dani, Orbits of horospherical flows, Duke Math. J. 53 (1986), no. 1, 177-188. MR 87i:22026
  • [DM] S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., vol. 16, Part 1, Amer. Math. Soc., Providence, RI, 1993, pp. 91-137. MR 95b:22024
  • [EMM] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), no. 1, 181-209. MR 95b:22025
  • [FJK] H. Fiedler, W. Jurkat, and O. Körner, Asymptotic expansions of finite theta series, Acta Arith. 32 (1977), no. 2, 129-146. MR 58:27832
  • [Ig] J. Igusa, On the graded ring of theta-constants. II, Amer. J. Math. 88 (1966), 221-236. MR 34:375
  • [HL1] G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation. II, Acta Math. 37 (1914), 193-239.
  • [HL2] -, A further note on the trigonometrical series associated with the elliptic theta-functions, Proc. Cambridge Philos. Soc. 21 (1922), 1-5.
  • [JVH1] W. B. Jurkat and J. W. Van Horne, The proof of the central limit theorem for theta sums, Duke Math. J. 48 (1981), no. 4, 873-885. MR 86m:11059
  • [JVH2] -, On the central limit theorem for theta series, Michigan Math. J. 29 (1982), no. 1, 65-77. MR 83h:10075
  • [JVH3] -, The uniform central limit theorem for theta sums, Duke Math. J. 50 (1983), no. 3, 649-666. MR 85g:11071
  • [Kli] H. Klingen, Introductory lectures on Siegel modular forms, Cambridge Stud. Adv. Math., vol. 20, Cambridge Univ. Press, Cambridge, 1990. MR 91a:11021
  • [LV] G. Lion and M. Vergne, The Weil representation, Maslov index and theta series, Progr. Math., vol. 6, Birkhäuser, Boston, MA, 1980. MR 81j:58075
  • [Ma1] J. Marklof, Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin, Emerging Applications of Number Theory (Minneapolis, MN, 1996), IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 405-450. MR 2000k:81094
  • [Ma2] -, Limit theorems for theta sums, Duke Math. J. 97 (1999), no. 1, 127-153. MR 2000h:11088
  • [Mo1] C. C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154-178. MR 33:1409
  • [Mo2] -, Exponential decay of correlation coefficients for geodesic flows, Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics (Berkeley, CA, 1984), Math. Sci. Res. Inst. Publ., vol. 6, Springer-Verlag, New York-Berlin, 1987, pp. 163-181. MR 89d:58102
  • [Mum] D. Mumford, Tata lectures on theta. I, Progr. Math., vol. 28, Birkhäuser, Boston, MA, 1983. MR 85h:14026
  • [R] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545-607. MR 93a:22009
  • [Sa] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math. 34 (1981), no. 6, 719-739. MR 83m:58060

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 11Fxx, 37D30

Retrieve articles in all journals with MSC (2000): 11Fxx, 37D30

Additional Information

F. Götze
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

M. Gordin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191011, Russia

Keywords: Theta series, Siegel's half-space, convergence in distribution, closed horospheres, unipotent flows
Received by editor(s): September 2, 2002
Published electronically: December 31, 2003
Additional Notes: Supported in part by the DFG-Forschergruppe FOR 399/1-1.
M. Gordin was also partially supported by RFBR (grant no. 02.01-00265) and by Sc. Schools grant no. 2258.2003.1. He was a guest of SFB-343 and the Department of Mathematics at the University of Bielefeld while the major part of this paper was prepared.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society